The Friendship Paradox | This is in your recommended because it relates to the spread of diseases

this video is sponsored by Coursera your friends on average are more popular than you now if you're like me your initial reaction to this is wait what friends okay maybe not but this is something that is true for a majority of people and it's not meant as an insult or anything like that but rather it's a mathematical fact that shows up in a lot of places for example years ago a study found that on Facebook the average person had 245 friends whereas the average friend of a person had 359 friends this here is the friendship paradox now it is called a paradox but really it's just mathematical truth that isn't obvious at first and real quick I want to note that this is not saying that most of your friends are more popular than you this is saying that if you lined up all your friends like let's say you have five friends and then you counted how many friends they each had or how popular they are and then average those numbers it would likely be higher than five the number of friends you have to simplify this let's imagine a world with four people in it that are connected like this where one of these connections means those two people are friends so yes these are reciprocal friendships in which both people that are connected would consider the other a friend and if any of the math you're about to see doesn't seem like it tells the full story I'm going to give some more detail on that after the sponsored portion at the end of this video I'm putting it there because it has to do with something more technical that doesn't take away from the math and applications we're about to see so let's continue now we see that Alice only has one friend or Bob in this case Bob on the other hand is the popular one who has three friends then the other two people have two friends each I'll put everything in a table as well to keep track of it all so to find how many friends the average person has we just add up those values and divide by the four people total leaving us with two friends on average so nothing strange yet but now we need to determine how popular people's friends are on average okay to avoid using the word friend a lot I'm going to use the word score to define how popular they are as in Bob would have a score of three so if we look at Alice she only has one friend or Bob in this case who has a score of three thus we can say alice's friends on average have a score of three this means her or friend in this case is more popular than her like the paradoxes for Bob his friends have a score of one two and two giving us an average of one point six seven so his friends actually aren't as popular as him which isn't too surprising then Carol's friends have scores of two and three leaving an average of two point five and for David we see the same thing his friends have a score of two and three which yields an average of two point five as we can see for most people the average score of their friends is higher than their own score or their friends are more popular the only exception is Bob which isn't surprising since he's the most popular now this does relate to what I said in the beginning but it's not exactly the full story we still want to find in general how popular the average friend is like with that Facebook example there were two numbers of interest we found for our example the average person has two friends but again we want to know how popular the average friend is so what does that even mean well we're not gonna average these values because that's not exactly what we're after that would just be averaging a bunch of averages instead just imagine everyone starts yelling out the name of each friend they have along with the corresponding score while someone else keeps track of those values so Alice would say Bob has the score of 3 which would be recorded Bob would say Alice has a score of 1 carol has a score of 2 and david has a score of 2 which are all also recorded Carol would say bob has a score of 3 and David has a score of 2 and then David would say Carol has a score of 2 and Bob has a score of 3 so everyone's just mentioning the popularity of each friend they have if we add everything up and divide by the eight numbers that were yelled out we get that the average friend has a score of 2.25 as you can see this is slightly more than the popularity of the average individual now I won't prove this but it turns out so long as everyone doesn't have the same number of friends this inequality will hold where the average friend will always be more popular than the average individual this is what the mathematics behind the friendship paradox really shows but now the question is why does this all happen well it's because you're more likely to be friends with someone that's more popular it's unlikely you're friends with someone very unpopular as they a few friends I mean there's only one bomb in this case one really popular person as most of us are not super popular but there are many who are friends with Bob and those that are now feel like they're relatively less popular because of who they're comparing themselves to also you notice before we are technically finding the weighted average as there were three 3s for Bob two twos for Carol another two for David and then a single one for Alice each individual score was weighted its own value and that weighted average will always exceed the individual average where everyone has counted just once so basically Bob is skewing the average more than anyone else both here and also for the averages up here an unpopular person on the other hand won't be in many friend groups so they don't have as much weight when it comes to calculating the popularity of the average friend plus from an individual viewpoint you wouldn't even have the chance to compare yourselves to them since they aren't a part of your group and that's probably not even on your radar now you may be wondering hey Zack is there any point to all of this or you're just trying to make a lot of us feel bad well actually the underlying analysis behind the friendship paradox shows up in our lives a lot more than you'd probably think so it's not just social media and friend groups in fact the most direct application has to do with preventing the spread of infectious diseases about a decade ago a study was conducted during a flu outbreak in which the health status of several undergrads was monitored what they did was first randomly select some people to keep track of but the clever thing they also did was ask those people to select a friend of theirs who was then also monitored they did this because chances are the friend that is chosen would be somewhat popular rather than someone with very few friends and a popular person is going to be more likely to catch the flu early on as well as spread it to more people due to their high number of connections these people are known as the super spreaders in a network the ones that are going to disproportionately infect others compared to unpopular students who have the disease as well and the study found exactly what was expected the group of selected friends caught the flu an average of two weeks earlier than the original people who are randomly selected these results show that we can use this method to catch a disease outbreak faster than traditional methods and it tells us that the randomly chosen friend would be prime targets for vaccination as that would have the biggest effect in slowing the spread of the disease so that was probably the most famous application but we also need to realize the friendship paradox is a form of sampling bias which is something that can easily make any of us come to very illogical conclusions on a day to day basis like when you compare your own popularity to that of your friends you're not comparing yourself to a truly representative sample of the population at least for most of you and that's because the people with very few friends are unlikely to appear in your friend groups while more popular people are and thus you have a biased sample see how the world really is versus what we experienced what we're exposed to sometimes aren't remotely the same like imagine a school that has only two classes offered and let's say one of those classes has 10 students in it and the other has 90 now looking at this we could say that the average class size is 50 and that's technically right but it's misleading because what would happen if we asked everyone what their experience with class size was we'd get that 90 students say that they're in a class of 90 and only 10 say they're in a class of 10 so out of the 100 students the weighted average or the average class size experienced by the students is 82 quite different from the 50 you see above see the average when we look at the classes as a whole does not reflect the average as experienced by the people within them although this isn't exactly the same it is related to the friendship paradox before we saw popular people are counted more often by their friends and here we see that the larger class size is mentioned more often during our survey it's all about weighted averages versus simple averages and when doing the analysis we need to remember that large classes or popular students carry more weight so when you're looking up information about that college you're considering don't let something like this fool you this examples for the University of Colorado and after doing the math while making some assumptions if you were to enroll in one class assuming all things random you'd have roughly a 21% chance of getting in one of those small classes a 36% chance of getting something in the middle and a 43% chance of being in a large class so what you see here is pretty much back from what you get and that's because small classes simply have less people in them and us you're less likely to be a part of one then something less mathematical but that still relates to sampling bias is the Jim even as an average person it can be easy to feel out of shape when you're surrounded by other gym goers but you have to remember who you're comparing yourself to it's the people who go to the gym often imagine the several people who just sit at home every day not working out you'll never see them at the gym so you don't even have the chance to compare yourself to them those even if your average you're often comparing yourself to people that are above average then there's a more political example of sampling bias that occurred in the presidential election between Franklin Roosevelt and Alf Landen in 1936 before the election literary digest magazine launched a huge survey to see if they could predict the outcome of the election what they did was sent out millions of mock ballots to people whose names were taken from telephone directories club membership lists magazine subscribers and so on this was all during the time of the Great Depression and thus who they were surveying was biased towards upper class and richer citizens that were more likely to vote Republican I mean remember this time having a phone was a luxury that not everyone could afford the survey returned that land and the Republican governor should win by a landslide when in reality the exact opposite happened but sampling bias goes much further than this like an observational astronomy for stars that are extremely far away only the brightest can be detected by our instruments or at least those above a certain threshold this bias in detecting only a specific set of stars can then create a false trend within the analysis of distance versus brightness since certain stars from very far away will never be detected there's Burton's paradox in which correlations that don't really exist can appear too due to certain biases and there's plenty more I'll be talking about in an upcoming video so as you can see making sense of the world we live in isn't always so easy there are sometimes there's things we don't think about or relationships that are hidden beneath the surface so in order for us to accurately make predictions or explain certain phenomena researchers often have to build complex models that can describe real-world scenarios for anyone who's interested in things like applied mathematics computer science economics and more the analysis of how these models are built will likely be very interesting and if you want to learn more about this Coursera offers a course called model thinking that will introduce you to the mathematics behind these models that describe our world model thinking is a very popular course on their site that not only takes you through some of the technical mathematics behind graph theory statistics game theory Markov processes and more but they'll use that information for the purpose of creating models that can be used to help us understand the world around us for example in relation to this video you'll learn about a process related to graph theory where new nodes introduced to a network have a higher probability of linking to those that already have a high score just like when someone moves to a new town and makes friends with an already popular person using this model we can see how a network will evolve in real time along with how everyone's degree or score will change over time the class will also take you through how random walks relate to stock market fluctuations and what we can learn from the drops and spikes that occur in the market you'll use Markov models to determine when a certain statistical equilibrium has been reached in a system that's constantly changing and there's plenty more you learn about in much more detail the course comes with built in lectures weekly quizzes and two exams I'll make sure you're up to date with everything plus there's a discussion forum where you can ask questions and discuss ideas with others who are taking the class so if you want to get started right now for free and also support the channel you can click the link in the description below ok now I waited until this part of the video to say this because I want you guys to see how the friendship paradox works according to most of the resources you'll find online along with all those applications without me having to say well technically this isn't necessarily true but and things like that however here I want to mention something that was really bugging me when I was doing the research for this video so the friendship paradox as proved by the math you see and almost any source you find online states that the average friend is more popular or just as popular as the average individual that is true it's what I showed an example of earlier and it's been proven but I send the beginning of for most people or more than half of you the average of your friend scores will be higher than yours and again we saw this was true in our example and I think this is how most people interpret the paradox but the thing is that phrasing of most of your friends on average are more popular than you isn't necessarily true in fact here's a counter example imagine now we have five people where three of them are friends with everyone thus they have four friends each then the other two are friends with everyone except each other thus they have three friends each if we look at this person here their friends have a score of three three four and four leaving an average of 3.5 which would be their average friends of friends value and the same thing could be said about the other two people who know everyone however for these people all of their friends have a score of four thus their average friends of friends score is also four so in this case three people are more popular than their friends whereas two are less popular this is exactly the opposite of what was mentioned before meaning it's not necessarily true that most people have friends that are more popular than them but know if everyone started yelling the name and score of each of their friends we'd still get that that average is higher than the average score of an individual because like I said that has been proven to always be the case but I'm just saying that's not the same thing as saying most of you are less popular than your friends so geez was this like all for nothing is this maybe a fallacy well no there is some good news although the friendship paradox as stated by most sources that you look at isn't necessarily true it is in fact usually true for the world that we live in the complexity of our friendship networks make it so yes most people's friends are on average more popular than them the mathematics behind this is much more complicated though so it's not meant for a video like this however that doesn't ruin the paradox and doesn't take away from any of the examples I gave so that's why I was putting this at the end of the video but with that we will end here if you guys enjoyed be sure to LIKE and subscribe follow me on Twitter and join the made Facebook group updates on everything hit the bell if you're not being notified and I'll see you all in the next video

Channel: Zach Star

Views: 909,886

Keywords: majorprep, major prep, the friendship paradox, friendship paradox, graph theory paradox, friendship network, graph theory, applications of graph theory, friendship paradox applications, paradox, counterintuitive math, sampling bias, applied math, applications of math, uses of math, applied graph theory, sampling bias examples, model thinking

Id: GEjhO65FYks

Length: 15min 33sec (933 seconds)

Published: Sun Jun 16 2019